Network motifs in cellular neurophysiology.

Concepts from network science and graph theory, including the framework of network motifs, have been frequently applied in studying neuronal networks and other biological complex systems. Network-based approaches can also be used to study the functions of individual neurons, where cellular elements such as ion channels and membrane voltage are conceptualized as nodes within a network, and their interactions are denoted by edges. Network motifs in this context provide functional building blocks that help to illuminate the principles of cellular neurophysiology. In this review we build a case that network motifs operating within neurons provide tools for defining the functional architecture of single-neuron physiology and neuronal adaptations. We highlight the presence of such computational motifs in the cellular mechanisms underlying action potential generation, neuronal oscillations, dendritic integration, and neuronal plasticity. Future work applying the network motifs perspective may help to decipher the functional complexities of neurons and their adaptation during health and disease.

Cross-strata co-occurrence of ripples with theta-frequency oscillations in the hippocampus of foraging rats.

Brain rhythms have been postulated to play central roles in animal cognition. A prominently reported dichotomy of hippocampal rhythms links theta-frequency oscillations (4-12 Hz) and ripples (120-250 Hz) exclusively to preparatory and consummatory behaviours, respectively. However, because of the differential power expression of these two signals across hippocampal strata, such exclusivity requires validation through analyses of simultaneous multi-strata recordings. We assessed co-occurrence of theta-frequency oscillations with ripples in multi-channel recordings of extracellular potentials across hippocampal strata from foraging rats. We detected all ripple events from an identified stratum pyramidale (SP) channel. We then defined theta epochs based on theta oscillations detected from the stratum lacunosum-moleculare (SLM) or the stratum radiatum (SR). We found ∼20% of ripple events (in SP) to co-occur with theta epochs identified from SR/SLM channels, defined here as theta ripples. Strikingly, when theta epochs were instead identified from the SP channel, such co-occurrences were significantly reduced because of a progressive reduction in theta power along the SLM-SR-SP axis. Behaviourally, we found most theta ripples to occur during immobile periods, with comparable theta power during exploratory and immobile theta epochs. Furthermore, the progressive reduction in theta power along the SLM-SR-SP axis was common to exploratory and immobile periods. Finally, we found a strong theta-phase preference of theta ripples within the fourth quadrant [3π/2 - 2π] of the associated theta oscillation. The prevalence of theta ripples expands the potential roles of ripple-frequency oscillations to span the continuum of encoding, retrieval and consolidation, achieved through interactions with theta oscillations. KEY POINTS: The brain manifests oscillations in recorded electrical potentials, with different frequencies of oscillation associated with distinct behavioural states. A prominently reported dichotomy assigns theta-frequency oscillations (4-12 Hz) and ripples (120-250 Hz) recorded in the hippocampus to be exclusively associated with preparatory and consummatory behaviours, respectively. Our multi-strata recordings from the rodent hippocampus coupled with cross-strata analyses provide direct quantitative evidence for the occurrence of ripple events nested within theta oscillations. These results highlight the need for an analysis pipeline that explicitly accounts for the specific strata where individual oscillatory power is high, in analysing simultaneously recorded data from multiple strata. Our observations open avenues for investigations involving cross-strata interactions between theta oscillations and ripples across different behavioural states.

The enigmatic HCN channels: A cellular neurophysiology perspective.

What physiological role does a slow hyperpolarization-activated ion channel with mixed cation selectivity play in the fast world of neuronal action potentials that are driven by depolarization? That puzzling question has piqued the curiosity of physiology enthusiasts about the hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, which are widely expressed across the body and especially in neurons. In this review, we emphasize the need to assess HCN channels from the perspective of how they respond to time-varying signals, while also accounting for their interactions with other co-expressing channels and receptors. First, we illustrate how the unique structural and functional characteristics of HCN channels allow them to mediate a slow negative feedback loop in the neurons that they express in. We present the several physiological implications of this negative feedback loop to neuronal response characteristics including neuronal gain, voltage sag and rebound, temporal summation, membrane potential resonance, inductive phase lead, spike triggered average, and coincidence detection. Next, we argue that the overall impact of HCN channels on neuronal physiology critically relies on their interactions with other co-expressing channels and receptors. Interactions with other channels allow HCN channels to mediate intrinsic oscillations, earning them the "pacemaker channel" moniker, and to regulate spike frequency adaptation, plateau potentials, neurotransmitter release from presynaptic terminals, and spike initiation at the axonal initial segment. We also explore the impact of spatially non-homogeneous subcellular distributions of HCN channels in different neuronal subtypes and their interactions with other channels and receptors. Finally, we discuss how plasticity in HCN channels is widely prevalent and can mediate different encoding, homeostatic, and neuroprotective functions in a neuron. In summary, we argue that HCN channels form an important class of channels that mediate a diversity of neuronal functions owing to their unique gating kinetics that made them a puzzle in the first place.

Heterogeneous off-target impact of ion-channel deletion on intrinsic properties of hippocampal model neurons that self-regulate calcium.

How do neurons that implement cell-autonomous self-regulation of calcium react to knockout of individual ion-channel conductances? To address this question, we used a heterogeneous population of 78 conductance-based models of hippocampal pyramidal neurons that maintained cell-autonomous calcium homeostasis while receiving theta-frequency inputs. At calcium steady-state, we individually deleted each of the 11 active ion-channel conductances from each model. We measured the acute impact of deleting each conductance (one at a time) by comparing intrinsic electrophysiological properties before and immediately after channel deletion. The acute impact of deleting individual conductances on physiological properties (including calcium homeostasis) was heterogeneous, depending on the property, the specific model, and the deleted channel. The underlying many-to-many mapping between ion channels and properties pointed to ion-channel degeneracy. Next, we allowed the other conductances (barring the deleted conductance) to evolve towards achieving calcium homeostasis during theta-frequency activity. When calcium homeostasis was perturbed by ion-channel deletion, post-knockout plasticity in other conductances ensured resilience of calcium homeostasis to ion-channel deletion. These results demonstrate degeneracy in calcium homeostasis, as calcium homeostasis in knockout models was implemented in the absence of a channel that was earlier involved in the homeostatic process. Importantly, in reacquiring homeostasis, ion-channel conductances and physiological properties underwent heterogenous plasticity (dependent on the model, the property, and the deleted channel), even introducing changes in properties that were not directly connected to the deleted channel. Together, post-knockout plasticity geared towards maintaining homeostasis introduced heterogenous off-target effects on several channels and properties, suggesting that extreme caution be exercised in interpreting experimental outcomes involving channel knockouts.

Plasticity manifolds and degeneracy govern circadian oscillations of neuronal intrinsic properties in the suprachiasmatic nucleus.

A heterogeneous neuronal population in the suprachiasmatic nucleus (SCN) sustains a cell-autonomous code for circadian time, implemented by firing-rate plasticity involving multiple ion channels. How do SCN neurons undergo stable firing-rate transitions if several ion channels change simultaneously in a heterogeneous neuronal population? Here, we addressed this question by building a heterogeneous population of SCN model neurons, each allowed to undergo one complete circadian cycle through multiple possible routes. We found that SCN neurons could achieve signature electrophysiological characteristics (day-like or night-like) despite pronounced heterogeneity in ion-channel conductances. Furthermore, for any neuron, disparate combinations of ion-channel plasticity yielded valid day-to-night or night-to-day transitions. Finally, nonlinear dimensionality reduction analyses on valid plasticity spaces revealed a low-dimensional plasticity manifold in day-to-night transitions, but not in night-to-day transitions. Our analyses unveil a synthesis of the degeneracy and the plasticity manifold frameworks that provides robustness and flexibility in achieving precise transitions despite widespread heterogeneities.

Degeneracy in epilepsy: multiple routes to hyperexcitable brain circuits and their repair.

Due to its complex and multifaceted nature, developing effective treatments for epilepsy is still a major challenge. To deal with this complexity we introduce the concept of degeneracy to the field of epilepsy research: the ability of disparate elements to cause an analogous function or malfunction. Here, we review examples of epilepsy-related degeneracy at multiple levels of brain organisation, ranging from the cellular to the network and systems level. Based on these insights, we outline new multiscale and population modelling approaches to disentangle the complex web of interactions underlying epilepsy and to design personalised multitarget therapies.

Ion-channel degeneracy and heterogeneities in the emergence of complex spike bursts in CA3 pyramidal neurons.

Complex spike bursting (CSB) is a characteristic electrophysiological signature exhibited by several neuronal subtypes and has been implicated in neural plasticity, learning, perception, anaesthesia and active sensing. Here, we address how pronounced intrinsic and synaptic heterogeneities affect CSB, with hippocampal CA3 pyramidal neurons (CA3PNs), where CSB emergence and heterogeneities are well characterized, as a substrate. We randomly generated 12,000 unique models and found 236 valid models that satisfied 11 characteristic CA3PN measurements. These morphologically and biophysically realistic valid models accounted for gating kinetics and somatodendritic expression profiles of 10 active ion channels. This heterogeneous population of valid models was endowed with broad distributions of underlying parameters showing weak pairwise correlations. We found two functional subclasses of valid models, intrinsically bursting and regular spiking, with significant differences in the expression of calcium and calcium-activated potassium conductances. We triggered CSB in all 236 models through different intrinsic or synaptic protocols and observed considerable heterogeneity in CSB propensity and properties spanning models and protocols. Finally, we used virtual knockout analyses and showed that synergistic interactions between intrinsic and synaptic mechanisms regulated CSB emergence and dynamics. Specifically, although there was a dominance of calcium and calcium-activated potassium channels in the emergence of CSB, individual deletion of none of the several ion channels or N-methyl-d-aspartate receptors resulted in the complete elimination of CSB across all models. Together, our analyses critically implicate ion-channel degeneracy in the robust emergence of CSB and other characteristic signatures of CA3PNs, despite pronounced heterogeneities in underlying intrinsic and synaptic properties. KEY POINTS: An unbiased stochastic search algorithm yielded a heterogeneous population of morphologically and biophysically realistic CA3 pyramidal neuronal models matching several signature electrophysiological characteristics. Two functional subclasses of valid models were identified with intrinsically bursting (IB) and regular spiking (RS) characteristics, which exhibited differential localization within the parametric space with linear and non-linear dimension reduction analyses. Calcium and calcium-activated potassium channels distinguished IB from RS models, apart from playing dominant roles in the emergence of complex spike bursting (CSB). The impact of deleting individual ion channels or N-methyl-d-aspartate receptors was variable across different models and differential for each channel/receptor, pointing to ion-channel degeneracy in the emergence of CSB. Biological heterogeneities across different neurons of the same subtype, ion-channel degeneracy and state-dependent changes (involving activity-dependent plasticity, pathology, and neuromodulation of intrinsic and synaptic properties) need to be considered carefully in assessing the propensity and dynamics of CSB in different neuronal subtypes.

Heterogeneous stochastic bifurcations explain intrinsic oscillatory patterns in entorhinal cortical stellate cells.

Stellate cells (SC) in the medial entorhinal cortex manifest intrinsic membrane potential oscillatory patterns. Although different theoretical frameworks have been proposed to explain these patterns, a robust unifying framework that jointly accounts for intrinsic heterogeneities and stochasticity is missing. Here, we first performed in vitro patch-clamp electrophysiological recordings from rat SCs and found pronounced cell-to-cell variability in their characteristic physiological properties, including peri-threshold oscillatory patterns. We demonstrate that noise introduced into two independent populations (endowed with deterministic or stochastic ion-channel gating kinetics) of heterogeneous biophysical models yielded activity patterns that were qualitatively similar to electrophysiological peri-threshold oscillatory activity in SCs. We developed spectrogram-based quantitative metrics for the identification of valid oscillations and confirmed that these metrics reliably captured the variable-amplitude and arhythmic oscillatory patterns observed in electrophysiological recordings. Using these quantitative metrics, we validated activity patterns from both heterogeneous populations of SC models, with each model assessed with multiple trials of different levels of noise at distinct membrane depolarizations. Our analyses unveiled the manifestation of stochastic resonance (detection of the highest number of valid oscillatory traces at an optimal level of noise) in both heterogeneous populations of SC models. Finally, we show that a generalized network motif comprised of a slow negative feedback loop amplified by a fast positive feedback loop manifested stochastic bifurcations and stochastic resonance in the emergence of oscillations. Together, through a unique convergence of the degeneracy and stochastic resonance frameworks, our unifying framework centered on heterogeneous stochastic bifurcations argues for state-dependent emergence of SC oscillations.

Efficient information coding and degeneracy in the nervous system.

Efficient information coding (EIC) is a universal biological framework rooted in the fundamental principle that system responses should match their natural stimulus statistics for maximizing environmental information. Quantitatively assessed through information theory, such adaptation to the environment occurs at all biological levels and timescales. The context dependence of environmental stimuli and the need for stable adaptations make EIC a daunting task. We argue that biological complexity is the principal architect that subserves deft execution of stable EIC. Complexity in a system is characterized by several functionally segregated subsystems that show a high degree of functional integration when they interact with each other. Complex biological systems manifest heterogeneities and degeneracy, wherein structurally different subsystems could interact to yield the same functional outcome. We argue that complex systems offer several choices that effectively implement EIC and homeostasis for each of the different contexts encountered by the system.

Dominant role of adult neurogenesis-induced structural heterogeneities in driving plasticity heterogeneity in dentate gyrus granule cells.

Neurons and synapses manifest pronounced variability in the amount of plasticity induced by identical activity patterns. The mechanisms underlying such plasticity heterogeneity, which have been implicated in context-specific resource allocation during encoding, have remained unexplored. Here, we employed a systematic physiologically constrained parametric search to identify the cellular mechanisms behind plasticity heterogeneity in dentate gyrus granule cells. We used heterogeneous model populations to ensure that our conclusions were not biased by parametric choices in a single hand-tuned model. We found that each of intrinsic, synaptic, and structural heterogeneities independently yielded heterogeneities in synaptic plasticity profiles obtained with two different induction protocols. However, among the disparate forms of neural-circuit heterogeneities, our analyses demonstrated the dominance of neurogenesis-induced structural heterogeneities in driving plasticity heterogeneity in granule cells. We found that strong relationships between neuronal intrinsic excitability and plasticity emerged only when adult neurogenesis-induced heterogeneities in neural structure were accounted for. Importantly, our analyses showed that it was not imperative that the manifestation of neural-circuit heterogeneities must translate to heterogeneities in plasticity profiles. Specifically, despite the expression of heterogeneities in structural, synaptic, and intrinsic neuronal properties, similar plasticity profiles were attainable across all models through synergistic interactions among these heterogeneities. We assessed the parametric combinations required for the manifestation of such degeneracy in the expression of plasticity profiles. We found that immature cells showed physiological plasticity profiles despite receiving afferent inputs with weak synaptic strengths. Thus, the high intrinsic excitability of immature granule cells was sufficient to counterbalance their low excitatory drive in the expression of plasticity profile degeneracy. Together, our analyses demonstrate that disparate forms of neural-circuit heterogeneities could mechanistically drive plasticity heterogeneity, but also caution against treating neural-circuit heterogeneities as proxies for plasticity heterogeneity. Our study emphasizes the need for quantitatively characterizing the relationship between neural-circuit and plasticity heterogeneities across brain regions.

Conjunctive changes in multiple ion channels mediate activity-dependent intrinsic plasticity in hippocampal granule cells.

Plasticity in the brain is ubiquitous. How do neurons and networks encode new information and simultaneously maintain homeostasis in the face of such ubiquitous plasticity? Here, we unveil a form of neuronal plasticity in rat hippocampal granule cells, which is mediated by conjunctive changes in HCN, inward-rectifier potassium, and persistent sodium channels induced by theta-modulated burst firing, a behaviorally relevant activity pattern. Cooperation and competition among these simultaneous changes resulted in a unique physiological signature: sub-threshold excitability and temporal summation were reduced without significant changes in action potential firing, together indicating a concurrent enhancement of supra-threshold excitability. This form of intrinsic plasticity was dependent on calcium influx through L-type calcium channels and inositol trisphosphate receptors. These observations demonstrate that although brain plasticity is ubiquitous, strong systemic constraints govern simultaneous plasticity in multiple components-referred here as plasticity manifolds-thereby providing a cellular substrate for concomitant encoding and homeostasis in engram cells.

Active Dendrites and Local Field Potentials: Biophysical Mechanisms and Computational Explorations.

Neurons and glial cells are endowed with membranes that express a rich repertoire of ion channels, transporters, and receptors. The constant flux of ions across the neuronal and glial membranes results in voltage fluctuations that can be recorded from the extracellular matrix. The high frequency components of this voltage signal contain information about the spiking activity, reflecting the output from the neurons surrounding the recording location. The low frequency components of the signal, referred to as the local field potential (LFP), have been traditionally thought to provide information about the synaptic inputs that impinge on the large dendritic trees of various neurons. In this review, we discuss recent computational and experimental studies pointing to a critical role of several active dendritic mechanisms that can influence the genesis and the location-dependent spectro-temporal dynamics of LFPs, spanning different brain regions. We strongly emphasize the need to account for the several fast and slow dendritic events and associated active mechanisms - including gradients in their expression profiles, inter- and intra-cellular spatio-temporal interactions spanning neurons and glia, heterogeneities and degeneracy across scales, neuromodulatory influences, and activitydependent plasticity - towards gaining important insights about the origins of LFP under different behavioral states in health and disease. We provide simple but essential guidelines on how to model LFPs taking into account these dendritic mechanisms, with detailed methodology on how to account for various heterogeneities and electrophysiological properties of neurons and synapses while studying LFPs.

Ion-channel degeneracy: Multiple ion channels heterogeneously regulate intrinsic physiology of rat hippocampal granule cells.

Degeneracy, the ability of multiple structural components to elicit the same characteristic functional properties, constitutes an elegant mechanism for achieving biological robustness. In this study, we sought electrophysiological signatures for the expression of ion-channel degeneracy in the emergence of intrinsic properties of rat hippocampal granule cells. We measured the impact of four different ion-channel subtypes-hyperpolarization-activated cyclic-nucleotide-gated (HCN), barium-sensitive inward rectifier potassium (Kir ), tertiapin-Q-sensitive inward rectifier potassium, and persistent sodium (NaP) channels-on 21 functional measurements employing pharmacological agents, and report electrophysiological data on two characteristic signatures for the expression of ion-channel degeneracy in granule cells. First, the blockade of a specific ion-channel subtype altered several, but not all, functional measurements. Furthermore, any given functional measurement was altered by the blockade of many, but not all, ion-channel subtypes. Second, the impact of blocking each ion-channel subtype manifested neuron-to-neuron variability in the quantum of changes in the electrophysiological measurements. Specifically, we found that blocking HCN or Ba-sensitive Kir channels enhanced action potential firing rate, but blockade of NaP channels reduced firing rate of granule cells. Subthreshold measures of granule cell intrinsic excitability (input resistance, temporal summation, and impedance amplitude) were enhanced by blockade of HCN or Ba-sensitive Kir channels, but were not significantly altered by NaP channel blockade. We confirmed that the HCN and Ba-sensitive Kir channels independently altered sub- and suprathreshold properties of granule cells through sequential application of pharmacological agents that blocked these channels. Finally, we found that none of the sub- or suprathreshold measurements of granule cells were significantly altered upon treatment with tertiapin-Q. Together, the heterogeneous many-to-many mapping between ion channels and single-neuron intrinsic properties emphasizes the need to account for ion-channel degeneracy in cellular- and network-scale physiology.

Stable continual learning through structured multiscale plasticity manifolds.

Biological plasticity is ubiquitous. How does the brain navigate this complex plasticity space, where any component can seemingly change, in adapting to an ever-changing environment? We build a systematic case that stable continuous learning is achieved by structured rules that enforce multiple, but not all, components to change together in specific directions. This rule-based low-dimensional plasticity manifold of permitted plasticity combinations emerges from cell type-specific molecular signaling and triggers cascading impacts that span multiple scales. These multiscale plasticity manifolds form the basis for behavioral learning and are dynamic entities that are altered by neuromodulation, metaplasticity, and pathology. We explore the strong links between heterogeneities, degeneracy, and plasticity manifolds and emphasize the need to incorporate plasticity manifolds into learning-theoretical frameworks and experimental designs.

Spatial information transfer in hippocampal place cells depends on trial-to-trial variability, symmetry of place-field firing, and biophysical heterogeneities.

The relationship between the feature-tuning curve and information transfer profile of individual neurons provides vital insights about neural encoding. However, the relationship between the spatial tuning curve and spatial information transfer of hippocampal place cells remains unexplored. Here, employing a stochastic search procedure spanning thousands of models, we arrived at 127 conductance-based place-cell models that exhibited signature electrophysiological characteristics and sharp spatial tuning, with parametric values that exhibited neither clustering nor strong pairwise correlations. We introduced trial-to-trial variability in responses and computed model tuning curves and information transfer profiles, using stimulus-specific (SSI) and mutual (MI) information metrics, across locations within the place field. We found spatial information transfer to be heterogeneous across models, but to reduce consistently with increasing levels of variability. Importantly, whereas reliable low-variability responses implied that maximal information transfer occurred at high-slope regions of the tuning curve, increase in variability resulted in maximal transfer occurring at the peak-firing location in a subset of models. Moreover, experience-dependent asymmetry in place-field firing introduced asymmetries in the information transfer computed through MI, but not SSI, and the impact of activity-dependent variability on information transfer was minimal compared to activity-independent variability. We unveiled ion-channel degeneracy in the regulation of spatial information transfer, and demonstrated critical roles for N-methyl-d-aspartate receptors, transient potassium and dendritic sodium channels in regulating information transfer. Our results demonstrate that trial-to-trial variability, tuning-curve shape and biological heterogeneities critically regulate the relationship between the spatial tuning curve and spatial information transfer in hippocampal place cells.

Resonating neurons stabilize heterogeneous grid-cell networks.

A central theme that governs the functional design of biological networks is their ability to sustain stable function despite widespread parametric variability. Here, we investigated the impact of distinct forms of biological heterogeneities on the stability of a two-dimensional continuous attractor network (CAN) implicated in grid-patterned activity generation. We show that increasing degrees of biological heterogeneities progressively disrupted the emergence of grid-patterned activity and resulted in progressively large perturbations in low-frequency neural activity. We postulated that targeted suppression of low-frequency perturbations could ameliorate heterogeneity-induced disruptions of grid-patterned activity. To test this, we introduced intrinsic resonance, a physiological mechanism to suppress low-frequency activity, either by adding an additional high-pass filter (phenomenological) or by incorporating a slow negative feedback loop (mechanistic) into our model neurons. Strikingly, CAN models with resonating neurons were resilient to the incorporation of heterogeneities and exhibited stable grid-patterned firing. We found CAN models with mechanistic resonators to be more effective in targeted suppression of low-frequency activity, with the slow kinetics of the negative feedback loop essential in stabilizing these networks. As low-frequency perturbations (1/f noise) are pervasive across biological systems, our analyses suggest a universal role for mechanisms that suppress low-frequency activity in stabilizing heterogeneous biological networks.

Ion-channel regulation of response decorrelation in a heterogeneous multi-scale model of the dentate gyrus.

Heterogeneities in biological neural circuits manifest in afferent connectivity as well as in local-circuit components such as neuronal excitability, neural structure and local synaptic strengths. The expression of adult neurogenesis in the dentate gyrus (DG) amplifies local-circuit heterogeneities and guides heterogeneities in afferent connectivity. How do neurons and their networks endowed with these distinct forms of heterogeneities respond to perturbations to individual ion channels, which are known to change under several physiological and pathophysiological conditions? We sequentially traversed the ion channels-neurons-network scales and assessed the impact of eliminating individual ion channels on conductance-based neuronal and network models endowed with disparate local-circuit and afferent heterogeneities. We found that many ion channels differentially contributed to specific neuronal or network measurements, and the elimination of any given ion channel altered several functional measurements. We then quantified the impact of ion-channel elimination on response decorrelation, a well-established metric to assess the ability of neurons in a network to convey complementary information, in DG networks endowed with different forms of heterogeneities. Notably, we found that networks constructed with structurally immature neurons exhibited functional robustness, manifesting as minimal changes in response decorrelation in the face of ion-channel elimination. Importantly, the average change in output correlation was dependent on the eliminated ion channel but invariant to input correlation. Our analyses suggest that neurogenesis-driven structural heterogeneities could assist the DG network in providing functional resilience to molecular perturbations.

Biomimetic FPGA-based spatial navigation model with grid cells and place cells.

The mammalian spatial navigation system is characterized by an initial divergence of internal representations, with disparate classes of neurons responding to distinct features including location, speed, borders and head direction; an ensuing convergence finally enables navigation and path integration. Here, we report the algorithmic and hardware implementation of biomimetic neural structures encompassing a feed-forward trimodular, multi-layer architecture representing grid-cell, place-cell and decoding modules for navigation. The grid-cell module comprised of neurons that fired in a grid-like pattern, and was built of distinct layers that constituted the dorsoventral span of the medial entorhinal cortex. Each layer was built as an independent continuous attractor network with distinct grid-field spatial scales. The place-cell module comprised of neurons that fired at one or few spatial locations, organized into different clusters based on convergent modular inputs from different grid-cell layers, replicating the gradient in place-field size along the hippocampal dorso-ventral axis. The decoding module, a two-layer neural network that constitutes the convergence of the divergent representations in preceding modules, received inputs from the place-cell module and provided specific coordinates of the navigating object. After vital design optimizations involving all modules, we implemented the tri-modular structure on Zynq Ultrascale+ field-programmable gate array silicon chip, and demonstrated its capacity in precisely estimating the navigational trajectory with minimal overall resource consumption involving a mere 2.92% Look Up Table utilization. Our implementation of a biomimetic, digital spatial navigation system is stable, reliable, reconfigurable, real-time with execution time of about 32 s for 100k input samples (in contrast to 40 minutes on Intel Core i7-7700 CPU with 8 cores clocking at 3.60 GHz) and thus can be deployed for autonomous-robotic navigation without requiring additional sensors.

Efficient phase coding in hippocampal place cells.

Neural codes have been postulated to build efficient representations of the external world. The hippocampus, an encoding system, employs neuronal firing rates and spike phases to encode external space. Although the biophysical origin of such codes is at a single neuronal level, the role of neural components in efficient coding is not understood. The complexity of this problem lies in the dimensionality of the parametric space encompassing neural components, and is amplified by the enormous biological heterogeneity observed in each parameter. A central question that spans encoding systems therefore is how neurons arrive at efficient codes in the face of widespread biological heterogeneities. To answer this, we developed a conductance-based spiking model for phase precession, a phase code of external space exhibited by hippocampal place cells. Our model accounted for several experimental observations on place cell firing and electrophysiology: the emergence of phase precession from exact spike timings of conductance-based models with neuron-specific ion channels and receptors; biological heterogeneities in neural components and excitability; the emergence of subthreshold voltage ramp, increased firing rate, enhanced theta power within the place field; a signature reduction in extracellular theta frequency compared to its intracellular counterpart; and experience-dependent asymmetry in firing-rate profile. We formulated phase-coding efficiency, using Shannon's information theory, as an information maximization problem with spike phase as the response and external space within a single place field as the stimulus. We employed an unbiased stochastic search spanning an 11-dimensional neural space, involving thousands of iterations that accounted for the biophysical richness and neuron-to-neuron heterogeneities. We found a small subset of models that exhibited efficient spatial information transfer through the phase code, and investigated the distinguishing features of this subpopulation at the parametric and functional scales. At the parametric scale, which spans the molecular components that defined the neuron, several nonunique parametric combinations with weak pairwise correlations yielded models with similar high phase-coding efficiency. Importantly, placing additional constraints on these models in terms of matching other aspects of hippocampal neural responses did not hamper parametric degeneracy. We provide quantitative evidence demonstrating this parametric degeneracy to be a consequence of a many-to-one relationship between the different parameters and phase-coding efficiency. At the functional scale, involving the cellular-scale neural properties, our analyses revealed an important higher-order constraint that was exclusive to models exhibiting efficient phase coding. Specifically, we found a counterbalancing negative correlation between neuronal gain and the strength of external synaptic inputs as a critical functional constraint for the emergence of efficient phase coding. These observations implicate intrinsic neural properties as important contributors in effectuating such counterbalance, which can be achieved by recruiting nonunique parametric combinations. Finally, we show that a change in afferent statistics, manifesting as input asymmetry onto these neuronal models, induced an adaptive shift in the phase code that preserved its efficiency. Together, our analyses unveil parametric degeneracy as a mechanism to harness widespread neuron-to-neuron heterogeneity towards accomplishing stable and efficient encoding, provided specific higher-order functional constraints on the relationship of neural gain to external inputs are satisfied.